## Monday Math 95

A notable arithmetic function which is neither additive nor multiplicative is the von Mangoldt function, denoted Λ(n). It is defined as

It has a few notable properties. First, consider . Considering the prime factorization , we see that the only divisors d of n for which the von Mangoldt function is non-zero are , , with . There are ki terms for each pi, so
,
or in terms of the Dirichlet convolution, (Λ*1)(n)=ln(n). Note that since Λ(1)=0, the von Mangoldt function has no Dirichlet inverse.

Now, let us consider a completely multiplicative function f(n), with Dirichlet inverse f-1(n), and Dirichlet series generating function . Now, let us consider the derivative of the Dirichlet series generating function. Taking the derivative with respect to s,
.

Next, let us consider the Dirichlet convolution of f-1(n) and f(n)ln(n). This is
.
For n=1, we note that ln(1)=0, so that the above is also zero in that case. Now, for n>1 with prime factorization , we have , with . I showed here that the Dirichlet inverse f-1(n) of a completely multiplicative function f(n) is the multiplicative function with
.
Thus, f-1(d)=0 whenever d is not squarefree, and so our terms are nonzero only when i1, i2, …, ir are each either 0 or 1; this reduces our sum to 2r terms. When all of the is are zero, so that d=1, the term in the sum is .
Now, suppose that only one of the is is zero, so that d is equal to one of the prime factors pj of n. Then
.
Thus, for those cases where n has only one distinct prime factor, , then
.
Next, for d a product of two distinct prime factors pj and pk,
.
So for n with , we see
.
And when d is the product of l distinct primes (2≤lr), , we see
.
Summing these, we see that an  can be factored from each term. Cancelling the logarithms, one finds that  whenever , so that the Dirichlet convolution is a function that is nonzero only for , p prime and k a positive integer; in that case, . Thus
.

Now, using the relationship between Dirichlet convolution and Dirichlet series, we have that the Dirichlet series generating function for  is thus the product of those for f-1(n) and f(n)ln(n); the former is , and we saw already that the latter is . Thus we see that the logarithmic derivative of F(s) has series
.
Letting the completely multiplicative function f(n) be 1(n), then we see

(for ), displaying the connection between the von Mangoldt function and the Riemann zeta function which is used in the earliest proofs of the prime number theorem.

Now, since Λ(1)=0, we see . Performing termwise integration with respect to s on the latter sum, one can then find
.